ON EIGENFUNCTIONS OF THE STAR GRAPHS Vladislav Kabanov Institute of Mathematics and Mechanics Yekaterinburg, Russia Joint work with S. Goryainov, E. Konstantinova, L. Shalaginov, and A. Valyuzhenich Shanghai Jao Tong University, March 14, 2019 On eigenfunctions of the Star Graphs
Outline • Eigenfunctions of graphs • Cayley graph C ay( G, S ) and Star graph S n • PI -eigenfunctions of the Star graph S n • A generalization of PI -eigenfunctions • Partitions and Young tableaux • Polytabloid eigenfunctions • Connection between polytabloid eigenfunctions and PI -eigenfunctions On eigenfunctions of the Star Graphs
Eigenfunctions of a graph Let Γ = ( V, E ) be a graph. A function f : V − → R is called an eigenfunction of the graph Γ corresponding to an eigenvalue θ if f �≡ 0 and for any vertex x the local condition � θ · f ( x ) = f ( y ) y ∈ Γ( x ) holds, where Γ( x ) is the neighbourhood of x . On eigenfunctions of the Star Graphs
Cayley graph and the neighbourhood of a vertex Let G be a finite group and S be a subset of G which does not contain the identity element and is closed under inversion. The Cayley graph of G generated by S , denoted by C ay( G, S ) , is the graph with: • vertex set G , • edge set { ( x, sx ) | for each x ∈ G, s ∈ S } . By the choice S the Cayley graph C ay( G, S ) is an undirected graph without loops and multiple edges. On eigenfunctions of the Star Graphs
Cayley graph and the neighbourhood of a vertex Let Γ be a Cayley graph C ay( G, S ) . For a given vertex x in Γ , let Γ( x ) = Sx = { sx | s ∈ S } . Then Γ( x ) is the neighbourhood of x in Γ . On eigenfunctions of the Star Graphs
Group algebra C [ G ] and the action by multiplication Let C be a complex field and let C [ G ] be the group algebra of G over C . For a subset S in G , consider the element S ∈ C [ G ] given by � S = s. s ∈ S Left multiplication of elements from C [ G ] by S is a linear transformation of C [ G ] . It is known the matrix of this linear transformation coincides with the adjacency matrix of C ay( G, S ) , hence spectral theory of Cayley graphs is connected with the theory of group representations. On eigenfunctions of the Star Graphs
The Star graph S n Let Ω be the set { 1 , . . . , n } , n ≥ 2 . We consider the symmetric group Sym Ω and put S = { (1 2) , (1 3) , . . . , (1 n ) } . The Star graph S n is the Cayley graph over the symmetric group Sym Ω with the generating set S . On eigenfunctions of the Star Graphs
The Star graph S n The Star graph is interesting as network topology, because it is an attractive alternative to the hypercube, a popular network for interconnecting processors in a parallel computer, and it compares favorably with hypercube in several aspects. In this talk we are interested in the eigenfunctions and spectrum of the Star graph S n . R. Ko’tter and F. R. Kschischang. Coding for errors and erasures in random network coding. IEEE Trans. Inf. Theory, 54, 8 (2008) 3579–3591. On eigenfunctions of the Star Graphs
The spectrum of the Star graph S n It was shown by Krakovski and Mohar that the spectrum of the Star graph S n contains all integers from − n + 1 to n − 1 (except 0 if n = 2 or n = 3 ). Since the Star graph is bipartite, mul( n − k ) = mul( − n + k ) for each integer 1 � k � n . Moreover, ± ( n − 1) are simple eigenvalues of S n . R. Krakovski, B. Mohar, Spectrum of Cayley graphs on the symmetric group generated by transposition, Linear Algebra and its Applications, 437 (2012) 1033–1039. On eigenfunctions of the Star Graphs
The spectrum of the Star graph S n In [1], Chapuy and Feray pointed out that the spectrum of the Star graph S n is integral, which was actually shown by Jucys in [2], who considered the action of the element ( n 1) + ( n 2) + . . . + ( n n − 1) on the complex group algebra over symmetric group C [ Sym n ] by left multiplication. However, eigenfunctions corresponding to these integral eigenvalues were never obtained explicitly. [1] G. Chapuy, V. Feray, A note on a Cayley graph of Sym n , arXiv:1202.4976v2 (2012) 1–3. [2] A. Jucys, Symmetric polynomials and the center of the symmetric group ring, Reports Math. Phys., 5 (1974) 107–112. On eigenfunctions of the Star Graphs
PI -eigenfunctions of the Star graph S n In this talk, we present a family of eigenfunctions of S n , n � 3 for any eigenvalue n − m − 1 , n > 2 m . S. Goryainov, V. V. Kabanov, E. Konstantinova, L. Shalaginov, A. Valyuzhenich, PI -eigenfunctions of the Star graphs, arXiv:1802.06611 (2018) 1–19. Submitted to Linear Algebra and its Applications On eigenfunctions of the Star Graphs
PI -eigenfunctions of the Star graph S n Let us define a m -tuple I m = ( i 1 , i 2 , . . . , i m ) of m pairwise distinct elements from the set { 1 , . . . , n } and a m -tuple P m = (( j 1 , k 1 ) , ( j 2 , k 2 ) , . . . , ( j m , k m )) , where 2 m pairwise distinct elements from the set { 2 , . . . , n } are arranged into m pairs. Define a function f P m I m : Sym n → R . For a permutation π = [ π 1 π 2 . . . π n ] ∈ Sym n , we put f P m I m ( π ) = 0 , if there exists t ∈ { 1 , 2 , . . . , m } such that π j t � = i t and π k t � = i t . If for every t ∈ { 1 , 2 , . . . , m } either π j t = i t or π k t = i t , then we define a binary vector X π = ( x 1 , x 2 , . . . , x m ) as follows: � 1 , if π j t = i t ; x t = 0 , if π k t = i t . We use the vector X π to complete the definition of the function f P m I m : On eigenfunctions of the Star Graphs
PI -eigenfunctions of the Star graph S n 1 , if wt ( X π ) is an even number; f P m I m ( π ) = − 1 , if wt ( X π ) is an odd number; 0 , if there exists t such that π j t � = i t and π k t � = i t . On eigenfunctions of the Star Graphs
PI -eigenfunctions of the Star graph S n 1 , if wt ( X π ) is an even number; f P m I m ( π ) = − 1 , if wt ( X π ) is an odd number; 0 , if there exists t such that π j t � = i t and π k t � = i t . Theorem 1. For n � 3 and n > 2 m > 0 the function f P m I m is an eigenfunction with eigenvalue n − m − 1 of the Star graph S n . We call the eigenfunction from Theorem 1 PI -eigenfunctions. On eigenfunctions of the Star Graphs
A generalization of PI -eigenfunctions Further we present a generalization of the family of PI -eigenfunctions of the Star graph S n , n � 3 , for all its non-zero eigenvalues. To present the generalized construction of eigenfunctions, we introduce some new definitions. On eigenfunctions of the Star Graphs
A generalization of PI -eigenfunctions Let us define two subsets Λ and ∆ of Ω with their partitions as follows. For any positive integers m and c such that m < n − 1 and c < n − m − 1 , let Λ be an m -element subset of Ω and Λ 1 , . . . , Λ c be its partition with parts of cardinality λ 1 , . . . , λ c , respectively. Thus, m = λ 1 + . . . + λ c . Denote this partition of m by λ = ( λ 1 , . . . , λ c ) . Let ∆ be an ( m + c ) -element subset of Ω \ { 1 } and ∆ 1 , . . . , ∆ c be its partition with parts of cardinality δ 1 , . . . , δ c , respectively, where δ t = λ t + 1 for each t ∈ { 1 , . . . , c } . Thus, m + c = δ 1 + . . . + δ c . Denote this partition of m + c by δ = ( δ 1 , . . . , δ c ) . On eigenfunctions of the Star Graphs
A generalization of PI -eigenfunctions Let V 1 and V 2 be m -ary and c -ary Cartesian powers of Ω , correspondingly, and V = V 1 × V 2 be their Cartesian product. For given Λ and ∆ , we take tuples I Λ ∈ V 1 , P ∆ ∈ V each of which contains only pairwise distinct elements of Ω . The m -tuple I Λ is given by the ordered parts ( I 1 , . . . , I c ) with respect to the partition λ of m such that I t = ( i t 1 , . . . , i tλ t ) , where i ts ∈ Λ t , 1 � s � λ t , for every t, 1 � t � c . The ( m + c ) -tuple P ∆ is given by the ordered parts ( P 1 , . . . , P c ) with respect to the partition δ of m + c such that P t = ( j t 1 , . . . , j tδ t ) , where j ts ∈ ∆ t , 1 � s � δ t , for every t, 1 � t � c . On eigenfunctions of the Star Graphs
A generalization of PI -eigenfunctions Let us consider the subgroup G of Sym Ω that fixes all symbols from Ω \ ∆ . Let H = { τ ∈ Sym Ω : τ (∆ t ) = ∆ t , 1 � t � c } be the stabilizer of the partition ∆ 1 , . . . , ∆ c , where τ (∆ t ) = ∆ t means that τ ( j ) ∈ ∆ t for any j ∈ ∆ t . For any π ∈ Sym Ω and τ ∈ H , by the relation I Λ ≺ π ( τ ( P ∆ )) , where I Λ ∈ V 1 and π ( τ ( P ∆ )) ∈ V , we mean that all entries of π ( τ ( P ∆ )) are the same as entries of I Λ at the corresponding positions, and π ( τ ( j tλ t )) / ∈ Λ t . On eigenfunctions of the Star Graphs
A generalization of PI -eigenfunctions We define the following function: +1 , if I Λ ≺ π ( τ ( P ∆ )) for even permutation τ ∈ H ; f P ∆ I Λ ( π ) = − 1 , if I Λ ≺ π ( τ ( P ∆ )) for odd permutation τ ∈ H ; 0 , if I Λ �≺ π ( τ ( P ∆ )) for any permutation τ ∈ H . (1) On eigenfunctions of the Star Graphs
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